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Additional info for Mathematics in Context: Patterns and Figures
2. a. Yes, both formulas are correct. Your explanation may differ from the ones presented here. If that is the case, discuss it with a classmate. Sample explanations: • If you fill in numbers 0, 1, 2, 3, etc. in both formulas, you get the same dot pattern. • 2(n ؉ 1) ؊ 1 ؍2n ؉ 2 ؊ 1 ؍2n ؉ 1. b. A table may help you find the answer for this problem. Pattern Number Number of Dots 1 1 2 3 3 5 4 7 5 9 6 11 David’s formula would change into W ؍2n ؊ 1 or W ؍2(n؉1)؊3. 3. a. You may have different patterns.
You could find the total number of tiles in different ways. For example, look at the pattern for the white tiles: Row Number from Base 1 2 3 4 5 6 7 Number of White Tiles 6 5 4 3 2 1 0 The total number of white tiles is 6 ؉ 5 ؉ 4 ؉ 3 ؉ 2 ؉ 1 ؍21. The total number of red tiles is 49 ؊ 21 ؍28. • You can look at the pattern of the total number of red triangles after each row. Row Number from Top 1 2 3 4 5 6 7 Total Red Tiles 1 3 6 10 15 21 28 ؉2 ؉3 ؉1 • 48 Patterns and Figures ؉4 ؉1 ؉5 ؉1 ؉6 ؉1 ؉7 ؉1 The total number of red tiles is 28, so the number of white tiles would be 49 ؊ 28 ؍21.
P؍1 P؍2 P؍4 P؍3 1. Write a direct formula to calculate the number of green tiles needed for each pattern number (P). Note that P starts at one in this problem! 2. Explain using the tile patterns above that the formula for the number of white tiles for pattern number P is number of white tiles ؍4 ؋ (P ؉ 1) Additional Practice 41 Additional Practice 3. What is the formula for the total number of tiles? 4. Explain how the formulas you found in problems 1, 2, and 3 are related. 5. Use an area diagram to show that the expressions n2 ؉ 4n ؉ 4 and (n ؉ 2)2 are equivalent.